$L^{p}-L^{q}$ existence for the open compressible MHD system

TL;DR

This paper establishes local existence of solutions for a compressible MHD system with inhomogeneous boundary conditions in the $L^p-L^q$ framework, addressing challenges posed by inflow boundaries and regularity estimates.

Contribution

It provides the first local well-posedness result for the full compressible MHD system with inflow boundary conditions in the $L^p-L^q$ class, using a linearization and fixed-point approach.

Findings

Proves local existence and uniqueness of solutions.

Handles inflow boundary conditions and density regularity issues.

Employs a linearization and Banach fixed-point theorem for proof.

Abstract

We study the local existence of solutions to the magnetohydrodynamics (MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the $L^p-L^q$ class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density $\varrho$ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of $\varrho$. The main result ensures the local well-posedness of the full MHD system which is shown through a linearization combined with a Banach fixed-point theorem.